Math242Final07 - 3-6 7 Convert the polar curve r = 4 sin θ...

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NAME MATH 242, Fall 2007 Final exam Arrange your work as clearly and neatly as possible, and cross out incorrect work. Unless otherwise noted, you must justify all answers to receive full credit. You may not use calculators, notes, or any other kinds of aids. Each question is worth 20 points, for a total of 200. 1. (a) Find dy dx if y = cosh ( 4 x ) . (b) Evaluate lim x tanh ( x ) x . d dx ( sin - 1 x ) = 1 1 - x 2 d dx ( tan - 1 x ) = 1 1 + x 2 1 - sin 2 θ = cos 2 θ 1 + tan 2 θ = sec 2 θ sin 2 θ = 1 2 ( 1 - cos 2 θ ) cos 2 θ = 1 2 ( 1 + cos 2 θ ) Logistic equation: dP dt = kP 1 - P K , P ( t ) = K 1 + Ae - kt , A = K - P 0 P 0 θ 0 π 6 π 4 π 3 π 2 2 π 3 3 π 4 5 π 6 π cos θ 1 3 2 1 2 1 2 0 - 1 2 - 1 2 - 3 2 - 1 sin θ 0 1 2 1 2 3 2 1 3 2 1 2 1 2 0
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2. Find e x 1 + e x dx .
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3. Evaluate lim x 0 tan ( 2 x 2 ) x 2 .
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4. Evaluate π / 4 0 4 sin 4 x dx , or show that it is divergent.
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5. Evaluate 1 0 ln ( x ) dx , or show that it is divergent.
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6. This question is about the curve x = t 3 - 3 t + 3, y = 2 t - 6. (a) Find equations for all of the vertical tangent lines. (b) Find the equation for the line tangent at the point
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Unformatted text preview: ( 3,-6 ) . 7. Convert the polar curve r = 4 sin θ to cartesian coordinates, and identify it as an ellipse, parabola, or hyperbola. 8. Find the Taylor series of f ( x ) = 1 ( x + 1 ) 2 at a = 0. 9. Determine whether ∞ ∑ n = 1 (-1 ) n n 2 n ! is absolutely convergent, conditionally convergent, or di-vergent. 10. A lake with a carrying capacity of 900 fish is stocked with 100 fish. The relative growth rate k is assumed to be equal to ln ( 2 ) per year. (a) How long will it take for the population to reach 300 fish? (Your answer should be simplified as far as possible.) (b) What would the answer to (a) be if the carrying capacity were essentially infinite?...
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